More generally, we deal with finite extensions of (functions on ), like (functions on an elliptic curve). And rings of -integers, like (poles only at ), (poles only at 0 and ).

Let be an absolutely almost simple -group with , like , , , for a quadratic form in at least 5 variables…. embeds in , completion of at place , as a non-uniform lattice. Let

It is the dimension of the Bruhat-Tits building of the product.

Theorem 1 (Bux-Wortman 2005, Bux-Köhl-Witzel 2011) is of type but not .

So, potentially, cohomology in degree might be infinite dimensional. It is indeed the case.

Theorem 2 (Stuhler 1980) If , is infinite dimensional for some finite index subgroup of .

I have a generalization.

Theorem 3 If the -type of is not , , or , then is infinite dimensional for some finite index subgroup of .

2. A simple example

Let us treat the abelian case: If , then is infinite. It is obvious, but I will give a fancy proof that is an introduction to general ideas. Take .

acts on a tree . Vertices are elements of . Put a common parent at height for two polynomials and if . Let polynomials of degree . It is a finite subgroup. Let polynomials of valuation at , so that . The quotient is a finite tree with an infinite ray attached to its root. Let be the vertices along the infinite ray issuing from the 0 polynomial. acts on . Let . It is -equivariant. Stuhler used equivariant cohomology to compute cohomology of , but the method gets unwieldy for other groups. Let us proceed differently, and introduce a different space , the full simplex with vertex set . It is simply connected with a free action of (so ), it comes with an equivariant map .

I describe explicit cocycles : , elsewhere. is -invariant, so it descends to . It is a cocycle. Let be an arc from 0 to in . Its image in is a cycle. On computes

so cohomology is infinite.

3. Now let us pass to

Proposition 4 is infinite.

Proof.

View as a subgroup of . Let be the Bruhat-Tits tree of that group. is a subtree there, it is a horoball. acts cocompactly on . View as the unipotent subgroup in . It acts on , and the induced action on is the same as before.

Let be the full simplex with vertex set . Average translates of cochain over cosets in to get a -invariant cochain . It descends to a 1-cocycle on . Again, its evaluation on shows cohomology is infinite.

Why does the theorem have exceptions ? The proof applies when (up to finite index), unipotent subgroups are abelian. This fails for , , …